Join Dr. William Murray in his Differential Equations online course complete with clear explanations of theory and a wide array of helpful insights. Each lesson also includes several step-by-step practice problems like the ones you will see on homework and tests.

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I. First-Order Equations

  Linear Equations 1:07:21
   Intro 0:00 
   Lesson Objectives 0:19 
   How to Solve Linear Equations 2:54 
    Calculate the Integrating Factor 2:58 
    Changes the Left Side so We Can Integrate Both Sides 3:27 
    Solving Linear Equations 5:32 
   Further Notes 6:10 
    If P(x) is Negative 6:26 
    Leave Off the Constant 9:38 
    The C Is Important When Integrating Both Sides of the Equation 9:55 
   Example 1 10:29 
   Example 2 22:56 
   Example 3 36:12 
   Example 4 39:24 
   Example 5 44:10 
   Example 6 56:42 
  Separable Equations 35:11
   Intro 0:00 
   Lesson Objectives 0:19 
    Some Equations Are Both Linear and Separable So You Can Use Either Technique to Solve Them 1:33 
    Important to Add C When You Do the Integration 2:27 
   Example 1 4:28 
   Example 2 10:45 
   Example 3 14:43 
   Example 4 19:21 
   Example 5 27:23 
  Slope & Direction Fields 1:11:36
   Intro 0:00 
   Lesson Objectives 0:20 
    If You Can Manipulate a Differential Equation Into a Certain Form, You Can Draw a Slope Field Also Known as a Direction Field 0:23 
    How You Do This 0:45 
   Solution Trajectories 2:49 
    Never Cross Each Other 3:44 
    General Solution to the Differential Equation 4:03 
    Use an Initial Condition to Find Which Solution Trajectory You Want 4:59 
   Example 1 6:52 
   Example 2 14:20 
   Example 3 26:36 
   Example 4 34:21 
   Example 5 46:09 
   Example 6 59:51 
  Applications, Modeling, & Word Problems of First-Order Equations 1:05:19
   Intro 0:00 
   Lesson Overview 0:38 
    Mixing 1:00 
    Population 2:49 
    Finance 3:22 
    Set Variables 4:39 
    Write Differential Equation 6:29 
    Solve It 10:54 
    Answer Questions 11:47 
   Example 1 13:29 
   Example 2 24:53 
   Example 3 32:13 
   Example 4 42:46 
   Example 5 55:05 
  Autonomous Equations & Phase Plane Analysis 1:01:20
   Intro 0:00 
   Lesson Overview 0:18 
    Autonomous Differential Equations Have the Form y' = f(x) 0:21 
    Phase Plane Analysis 0:48 
    y' < 0 2:56 
    y' > 0 3:04 
    If we Perturb the Equilibrium Solutions 5:51 
    Equilibrium Solutions 7:44 
    Solutions Will Return to Stable Equilibria 8:06 
    Solutions Will Tend Away From Unstable Equilibria 9:32 
    Semistable Equilibria 10:59 
   Example 1 11:43 
   Example 2 15:50 
   Example 3 28:27 
   Example 4 31:35 
   Example 5 43:03 
   Example 6 49:01 

II. Second-Order Equations

  Distinct Roots of Second Order Equations 28:44
   Intro 0:00 
   Lesson Overview 0:36 
    Linear Means 0:50 
    Second-Order 1:15 
    Homogeneous 1:30 
    Constant Coefficient 1:55 
    Solve the Characteristic Equation 2:33 
    Roots r1 and r2 3:43 
    To Find c1 and c2, Use Initial Conditions 4:50 
   Example 1 5:46 
   Example 2 8:20 
   Example 3 16:20 
   Example 4 18:26 
   Example 5 23:52 
  Complex Roots of Second Order Equations 31:49
   Intro 0:00 
   Lesson Overview 0:15 
    Sometimes The Characteristic Equation Has Complex Roots 1:12 
   Example 1 3:21 
   Example 2 7:42 
   Example 3 15:25 
   Example 4 18:59 
   Example 5 27:52 
  Repeated Roots & Reduction of Order 43:02
   Intro 0:00 
   Lesson Overview 0:23 
    If the Characteristic Equation Has a Double Root 1:46 
    Reduction of Order 3:10 
   Example 1 7:23 
   Example 2 9:20 
   Example 3 14:12 
   Example 4 31:49 
   Example 5 33:21 
  Undetermined Coefficients of Inhomogeneous Equations 50:01
   Intro 0:00 
   Lesson Overview 0:11 
    Inhomogeneous Equation Means the Right Hand Side is Not 0 Anymore 0:21 
    First Solve the Inhomogeneous Equation 1:04 
    Find a Particular Solution to the Inhomogeneous Equation Using Undetermined Coefficients 2:03 
    g(t) vs. Guess for ypar 2:42 
    If Any Term of Your Guess for ypar Looks Like Any Term of yhom 5:07 
   Example 1 7:54 
   Example 2 15:25 
   Example 3 23:45 
   Example 4 33:35 
   Example 5 42:57 
  Inhomogeneous Equations: Variation of Parameters 49:22
   Intro 0:00 
   Lesson Overview 0:31 
    Inhomogeneous vs. Homogeneous 0:47 
    First Solve the Inhomogeneous Equation 1:17 
    Notice There is No Coefficient in Front of y'' 1:27 
    Find a Particular Solution to the Inhomogeneous Equation Using Variation of Parameters 2:32 
    How to Solve 4:33 
    Hint on Solving the System 5:23 
   Example 1 7:27 
   Example 2 17:46 
   Example 3 23:14 
   Example 4 31:49 
   Example 5 36:00 

III. Series Solutions

  Review of Power Series 57:38
   Intro 0:00 
   Lesson Overview 0:36 
    Taylor Series Expansion 0:37 
    Maclaurin Series 2:36 
    Common Maclaurin Series to Remember From Calculus 3:35 
    Radius of Convergence 7:58 
    Ratio Test 12:05 
   Example 1 15:18 
   Example 2 20:02 
   Example 3 27:32 
   Example 4 39:33 
   Example 5 45:42 
  Series Solutions Near an Ordinary Point 1:20:28
   Intro 0:00 
   Lesson Overview 0:49 
    Guess a Power Series Solution and Calculate Its Derivatives, Example 1 1:03 
    Guess a Power Series Solution and Calculate Its Derivatives, Example 2 3:14 
    Combine the Series 5:00 
    Match Exponents on x By Shifting Indices 5:11 
    Match Starting Indices By Pulling Out Initial Terms 5:51 
    Find a Recurrence Relation on the Coefficients 7:09 
   Example 1 7:46 
   Example 2 19:10 
   Example 3 29:57 
   Example 4 41:46 
   Example 5 57:23 
   Example 6 69:12 
  Euler Equations 24:42
   Intro 0:00 
   Lesson Overview 0:11 
    Euler Equation 0:15 
    Real, Distinct Roots 2:22 
    Real, Repeated Roots 2:37 
    Complex Roots 2:49 
   Example 1 3:51 
   Example 2 6:20 
   Example 3 8:27 
   Example 4 13:04 
   Example 5 15:31 
   Example 6 18:31 
  Series Solutions 1:20:26
   Intro 0:00 
   Lesson Overview 0:13 
    Singular Point 1:17 
   Definition: Pole of Order n 1:58 
    Pole Of Order n 2:04 
    Regular Singular Point 3:25 
   Solving Around Regular Singular Points 7:08 
    Indical Equation 7:30 
    If the Difference Between the Roots is An Integer 8:06 
    If the Difference Between the Roots is Not An Integer 8:29 
   Example 1 8:47 
   Example 2 14:57 
   Example 3 25:40 
   Example 4 47:23 
   Example 5 69:01 

IV. Laplace Transform

  Laplace Transforms 41:52
   Intro 0:00 
   Lesson Overview 0:09 
    Laplace Transform of a Function f(t) 0:18 
    Laplace Transform is Linear 1:04 
   Example 1 1:43 
   Example 2 18:30 
   Example 3 22:06 
   Example 4 28:27 
   Example 5 33:54 
  Inverse Laplace Transforms 47:05
   Intro 0:00 
   Lesson Overview 0:09 
    Laplace Transform L{f} 0:13 
    Run Partial Fractions 0:24 
   Common Laplace Transforms 1:20 
   Example 1 3:24 
   Example 2 9:55 
   Example 3 14:49 
   Example 4 22:03 
   Example 5 33:51 
  Laplace Transform Initial Value Problems 45:15
   Intro 0:00 
   Lesson Overview 0:12 
    Start With Initial Value Problem 0:14 
    Take the Laplace Transform of Both Sides of the Differential Equation 0:37 
    Plug in the Identities 1:20 
    Take the Inverse Laplace Transform to Find y 2:40 
    Example 1 4:15 
    Example 2 11:30 
    Example 3 17:59 
    Example 4 24:51 
    Example 5 36:05 

V. Review of Linear Algebra

  Review of Linear Algebra 57:30
   Intro 0:00 
   Lesson Overview 0:41 
    Matrix 0:54 
    Determinants 4:45 
   3x3 Determinants 5:08 
   Eigenvalues and Eigenvectors 7:01 
    Eigenvector 7:48 
    Eigenvalue 7:54 
   Lesson Overview 8:17 
    Characteristic Polynomial 8:47 
    Find Corresponding Eigenvector 9:03 
   Example 1 10:19 
   Example 2 16:49 
   Example 3 20:52 
   Example 4 25:34 
   Example 5 35:05 

VI. Systems of Equations

  Distinct Real Eigenvalues 59:26
   Intro 0:00 
   Lesson Overview 1:11 
   How to Solve Systems 2:48 
    Find the Eigenvalues and Their Corresponding Eigenvectors 2:50 
    General Solution 4:30 
    Use Initial Conditions to Find c1 and c2 4:57 
   Graphing the Solutions 5:20 
    Solution Trajectories Tend Towards 0 or ∞ Depending on Whether r1 or r2 are Positive or Negative 6:35 
    Solution Trajectories Tend Towards the Axis Spanned by the Eigenvector Corresponding to the Larger Eigenvalue 7:27 
   Example 1 9:05 
   Example 2 21:06 
   Example 3 26:38 
   Example 4 36:40 
   Example 5 43:26 
   Example 6 51:33 
  Complex Eigenvalues 1:03:54
   Intro 0:00 
   Lesson Overview 0:47 
    Recall That to Solve the System of Linear Differential Equations, We find the Eigenvalues and Eigenvectors 0:52 
    If the Eigenvalues are Complex, Then They Will Occur in Conjugate Pairs 1:13 
   Expanding Complex Solutions 2:55 
    Euler's Formula 2:56 
    Multiply This Into the Eigenvector, and Separate Into Real and Imaginary Parts 1:18 
   Graphing Solutions From Complex Eigenvalues 5:34 
   Example 1 9:03 
   Example 2 20:48 
   Example 3 28:34 
   Example 4 41:28 
   Example 5 51:21 
  Repeated Eigenvalues 45:17
   Intro 0:00 
   Lesson Overview 0:44 
    If the Characteristic Equation Has a Repeated Root, Then We First Find the Corresponding Eigenvector 1:14 
    Find the Generalized Eigenvector 1:25 
   Solutions from Repeated Eigenvalues 2:22 
    Form the Two Principal Solutions and the Two General Solution 2:23 
    Use Initial Conditions to Solve for c1 and c2 3:41 
   Graphing the Solutions 3:53 
   Example 1 8:10 
   Example 2 16:24 
   Example 3 23:25 
   Example 4 31:04 
   Example 5 38:17 

VII. Inhomogeneous Systems

  Undetermined Coefficients for Inhomogeneous Systems 43:37
   Intro 0:00 
   Lesson Overview 0:35 
    First Solve the Corresponding Homogeneous System x'=Ax 0:37 
   Solving the Inhomogeneous System 2:32 
    Look for a Single Particular Solution xpar to the Inhomogeneous System 2:36 
    Plug the Guess Into the System and Solve for the Coefficients 3:27 
    Add the Homogeneous Solution and the Particular Solution to Get the General Solution 3:52 
   Example 1 4:49 
   Example 2 9:30 
   Example 3 15:54 
   Example 4 20:39 
   Example 5 29:43 
   Example 6 37:41 
  Variation of Parameters for Inhomogeneous Systems 1:08:12
   Intro 0:00 
   Lesson Overview 0:37 
    Find Two Solutions to the Homogeneous System 2:04 
    Look for a Single Particular Solution xpar to the inhomogeneous system as follows 2:59 
   Solutions by Variation of Parameters 3:35 
   General Solution and Matrix Inversion 6:35 
    General Solution 6:41 
    Hint for Finding Ψ-1 6:58 
   Example 1 8:13 
   Example 2 16:23 
   Example 3 32:23 
   Example 4 37:34 
   Example 5 49:00 

VIII. Numerical Techniques

  Euler's Method 45:30
   Intro 0:00 
   Lesson Overview 0:32 
    Euler's Method is a Way to Find Numerical Approximations for Initial Value Problems That We Cannot Solve Analytically 0:34 
    Based on Drawing Lines Along Slopes in a Direction Field 1:18 
   Formulas for Euler's Method 1:57 
   Example 1 4:47 
   Example 2 14:45 
   Example 3 24:03 
   Example 4 33:01 
   Example 5 37:55 
  Runge-Kutta & The Improved Euler Method 41:04
   Intro 0:00 
   Lesson Overview 0:43 
    Runge-Kutta is Know as the Improved Euler Method 0:46 
    More Sophisticated Than Euler's Method 1:09 
    It is the Fundamental Algorithm Used in Most Professional Software to Solve Differential Equations 1:16 
    Order 2 Runge-Kutta Algorithm 1:45 
   Runge-Kutta Order 2 Algorithm 2:09 
   Example 1 4:57 
   Example 2 10:57 
   Example 3 19:45 
   Example 4 24:35 
   Example 5 31:39 

IX. Partial Differential Equations

  Review of Partial Derivatives 38:22
   Intro 0:00 
   Lesson Overview 1:04 
    Partial Derivative of u with respect to x 1:37 
    Geometrically, ux Represents the Slope As You Walk in the x-direction on the Surface 2:47 
   Computing Partial Derivatives 3:46 
    Algebraically, to Find ux You Treat The Other Variable t as a Constant and Take the Derivative with Respect to x 3:49 
    Second Partial Derivatives 4:16 
    Clairaut's Theorem Says that the Two 'Mixed Partials' Are Always Equal 5:21 
   Example 1 5:34 
   Example 2 7:40 
   Example 3 11:17 
   Example 4 14:23 
   Example 5 31:55 
  The Heat Equation 44:40
   Intro 0:00 
   Lesson Overview 0:28 
    Partial Differential Equation 0:33 
    Most Common Ones 1:17 
    Boundary Value Problem 1:41 
   Common Partial Differential Equations 3:41 
    Heat Equation 4:04 
    Wave Equation 5:44 
    Laplace's Equation 7:50 
   Example 1 8:35 
   Example 2 14:21 
   Example 3 21:04 
   Example 4 25:54 
   Example 5 35:12 
  Separation of Variables 57:44
   Intro 0:00 
   Lesson Overview 0:26 
    Separation of Variables is a Technique for Solving Some Partial Differential Equations 0:29 
   Separation of Variables 2:35 
    Try to Separate the Variables 2:38 
    If You Can, Then Both Sides Must Be Constant 2:52 
    Reorganize These Intro Two Ordinary Differential Equations 3:05 
   Example 1 4:41 
   Example 2 11:06 
   Example 3 18:30 
   Example 4 25:49 
   Example 5 32:53 
  Fourier Series 1:24:33
   Intro 0:00 
   Lesson Overview 0:38 
    Fourier Series 0:42 
    Find the Fourier Coefficients by the Formulas 2:05 
   Notes on Fourier Series 3:34 
    Formula Simplifies 3:35 
    Function Must be Periodic 4:23 
   Even and Odd Functions 5:37 
    Definition 5:45 
    Examples 6:03 
   Even and Odd Functions and Fourier Series 9:47 
    If f is Even 9:52 
    If f is Odd 11:29 
   Extending Functions 12:46 
    If We Want a Cosine Series 14:13 
    If We Wants a Sine Series 15:20 
   Example 1 17:39 
   Example 2 43:23 
   Example 3 51:14 
   Example 4 61:52 
   Example 5 71:53 
  Solution of the Heat Equation 47:41
   Intro 0:00 
   Lesson Overview 0:22 
   Solving the Heat Equation 1:03 
   Procedure for the Heat Equation 3:29 
    Extend So That its Fourier Series Will Have Only Sines 3:57 
    Find the Fourier Series for f(x) 4:19 
   Example 1 5:21 
   Example 2 8:08 
   Example 3 17:42 
   Example 4 25:13 
   Example 5 28:53 
   Example 6 42:22 

Duration: 26 hours, 33 minutes

Number of Lessons: 30

This course is perfect for the college student taking Differential Equations and will help you understand & solve problems from all over biology, physics, chemistry, and engineering. No more getting stuck in one of the hardest college math courses.

Additional Features:

  • Free Sample Lessons
  • Closed Captioning (CC)
  • Practice Questions
  • Downloadable Lecture Slides
  • Study Guides
  • Instructor Comments

Topics Include:

  • Separable Equations
  • Direction Fields
  • Second Order Equations
  • Euler Equations
  • Laplace Transforms
  • Eigenvalues & Eigenvectors
  • Inhomogenous Systems
  • Partial Derivatives
  • Heat Equation
  • Fourier Series

Dr. Murray received his Ph.D from UC Berkeley, his BS from Georgetown University, and has been teaching in the university setting for 15+ years.

Student Testimonials:

"I'm very glad there are still great professors out there such as yourself who know exactly how to teach. I'm very grateful."  — Mateusz M.

“Hi Dr. Murray, your lectures are great and clear.” — Kyung Yeop K.

“Excellent! Wunderbar!” — William D.

“Great teaching professor. I have learned so much from your videos. I cannot thank you enough.” — Eddy N.

“Let me first say, thanks so much for making these as quickly as you did, I really didn't expect they'd be complete until after my DE course. These are fantastic, I'm very grateful.” — Caleb L.